Inverse problems based on partial differential equations and integral equations

Transcription

1 Inverse problems based on partial differential equations and integral equations Jijun Liu Department of Mathematics, Southeast University Korea, November 22, 214 Jijun Liu Inverse problems based on partial differential equations and integral equations 1 / 72

2 Outline 1 Obstacle imaging based on Helmholtz equation model Impedance boundary conditions Mixed boundary conditions Oblique derivative conditions 2 Image deblurring based on fractional diffusion model Backward diffusion model for image blurring Regularizing scheme by TV penalty Reconstruction scheme 3 Combustion control by absorbtion spectroscopy model Temperature and concentration detection in combustion Parameters detections by nonlinear integral equation Semilinear alternative iteration schemes Jijun Liu Inverse problems based on partial differential equations and integral equations 2 / 72

3 Obstacle imaging based on Helmholtz equation model Introduction to scattering problems Scattering theory is concerned with the interaction of acoustic, electromagnetic or elastic waves on the obstacles and inhomogeneities. Direct problem is to find scattered waves, if we are given the scatterer and incident wave. Inverse problem aims to determine the scatterer from a knowledge of the incident wave and the measured scattering data. Jijun Liu Inverse problems based on partial differential equations and integral equations 3 / 72

4 Obstacle imaging based on Helmholtz equation model Scattering configuration Figure 1.1. Scattering of u i by an obstacle D Jijun Liu Inverse problems based on partial differential equations and integral equations 4 / 72

5 Obstacle imaging based on Helmholtz equation model Scattering by impenetrable obstacles u + k 2 u = in R n \ D, Helmholtz equation B[u](x) = on D, Boundary condition ( lim 2 u s r iku s) =. Sommerfeld radiation condition, r r n 1 u = u i + u s : total wave, u s : scattered wave outside of D. B represents the physical property of obstacle D. Four different forms of operator B: B[u] = u, (Dirichlet) B[u] = u + iλu, (Impedance/Neumann) ν B[u] = u ν + iλ u, (Oblique derivative). τ There may be mixed boundary condition on D. Jijun Liu Inverse problems based on partial differential equations and integral equations 5 / 72

6 Obstacle imaging based on Helmholtz equation model Inverse scattering problems The scattered wave u s (, d) has the asymptotic behavior as x : ( ( )) u s (x, d) = eik x 1 u (ˆx, d) + O, ˆx := x x x x (n 1)/2 u (ˆx, d): far-field pattern of u s (x, d) corresponding to incident direction d, which is physically measurable. Inverse problems: Reconstruct the information of obstacle boundary D from given u i and u such as Geometric shape D; Boundary type (Dirichlet, Neumann, Impedance, Oblique derivative); Distinguish different parts of the boundary; Boundary parameters: λ(x). Jijun Liu Inverse problems based on partial differential equations and integral equations 6 / 72

7 Obstacle imaging based on Helmholtz equation model Inverse scattering by impedance obstacles u + k 2 u = in R n \ D, Helmholtz equation u ν + iλu = on D, Boundary condition ( lim 2 u s r iku s) =. Sommerfeld radiation condition r r n 1 Impedance boundary condition can be used to describe some scattering processes involving complex geometric structures or physical properties, such as scattering from an imperfectly conductor; scattering from a coated obstacle. Inverse problem: Reconstruct D or (and) λ for given u i and u. Jijun Liu Inverse problems based on partial differential equations and integral equations 7 / 72

8 Obstacle imaging based on Helmholtz equation model Detecting D for unknown impedance λ(x) by non-iterative methods Key idea of non-iterative methods: Construct an indicator function I (z) using far-field data, which blows up as z approaches to D. Remarks: Don t need any a priori information about the unknown scatterer; It is non-iterative; Essences of these methods lie in the reconstruction of the Green function associated with the boundary value problems for the Helmholtz equation. Jijun Liu Inverse problems based on partial differential equations and integral equations 8 / 72

9 Obstacle imaging based on Helmholtz equation model Detecting D for unknown impedance λ(x) by non-iterative methods Researches in this area by our group:. Liu Jijun, Cheng Jin, G. Nakamura, Science in China, 45(11), 22. J.Cheng, J.J.Liu, G.Nakamura, Inverse Problems, 21(3), 25. F.B.Hassen, J.J.Liu, R.Potthast, J. Comput. Maths, 25(3), 27. J.J.Liu, G.Nakamura, M.Sini, SIAM J. Appl. Maths, 67(4), 27. J.J.Liu, M.Sini, SIAM J. Scientific Computing, 31(4), 29. H.B.Wang, J.J.Liu, Science China Mathematics, 53(8), 21. H.B.Wang, J.J.Liu, Advances Comput. Maths, 36(2), 212. J.J.Liu, H.F.Zhao, Inverse Problems Sci. Engin., 21(4), 213. H.B. Wang, J.J. Liu, Applicable Analysis, 92(4), 213. Y.Q.Hu, F.Cakoni, J.J.Liu, Applicable Analysis, 214. F.Cakoni, Y.Q.Hu, R.Kress, Inverse Problems Jijun Liu Inverse problems based on partial differential equations and integral equations 9 / 72

10 Obstacle imaging based on Helmholtz equation model Problem: Reconstruct the impedance from the far-field data provided D is known Applications: Antenna design: We can obtain certain radiation pattern by suitably choosing the surface impedance; Detection of decoys: The hostile decoy in practice can become a perfect conductor coated by a thin dielectric layer. The shapes of the real object and decoy are the same and known in advance. The surface impedance serves as a target signature. Jijun Liu Inverse problems based on partial differential equations and integral equations 1 / 72

11 Obstacle imaging based on Helmholtz equation model Recovery of the impedance from the far-field data Step 1: Construct scattered wave and its far-field patterns of point source from far-field patterns of incident plane waves Φ(, ) : fundamental solution Φ s (, z) : scattered field for point source Φ(, z) Φ (, z) : far-field pattern of Φ s (, z) u u s Φ Φ s (x, z). : reconstruct near field from its far-field : mixed reciprocity principle Jijun Liu Inverse problems based on partial differential equations and integral equations 11 / 72

12 Obstacle imaging based on Helmholtz equation model Recovery of the impedance from the far-field data Step 2: Construct density functions for the Herglotz wave function to approximate the point source For any domain B containing D, and z B, we take a sequence {g n (, z)} n=1 in L2 (S) such that H[g n ](, z) Φ(, z) L 2 ( B), n, (1) where H is the Herglotz wave operator defined by H[p](x) := e ikx d p(d)ds(d). It can be deduced from (1) that S H[g n ](, z) Φ(, z) H 1 (D), n. (2) Jijun Liu Inverse problems based on partial differential equations and integral equations 12 / 72

13 Obstacle imaging based on Helmholtz equation model Recovery of the impedance by integral equation method Step 3: Establish the integral equation of the first kind for λ(x) Theorem For z j R 2 \ D, x R 2 \ D, x z j, define G(x, z j ) = Φ(x, z j ) + Φ s (x, z j ), j = 1, 2. Then we have λ(x)g(x, z 1 )G(x, z 2 )ds(x) D = k Φ (d, z 1 )Φ (d, z 2 )ds(d) i S 2 lim 1 Φ (d, z 1 )g n (d, z 2 )ds(d) n σ S + i 2 lim 1 Φ n σ (d, z 2 )g n (d, z 1 )ds(d) S := F(z 1, z 2 ). (3) This equation is ill-posed. Can we recover λ using (3)? Jijun Liu Inverse problems based on partial differential equations and integral equations 13 / 72

14 Obstacle imaging based on Helmholtz equation model Recovery of the impedance by integral equation method Theorem Define the operator A by A[λ](z 1, z 2 ) := λ(x)g(x, z 1 )G(x, z 2 )ds(x) (4) D for z 1, z 2 Ω, where Ω is the boundary of a bounded domain containing D in its interior. Then A is an injective compact integral operator from L 2 ( D) to L 2 (Ω Ω). Then, we can determine λ from A[λ](z 1, z 2 ) = F(z 1, z 2 ) by using the Tikhonov regularization method. Jijun Liu Inverse problems based on partial differential equations and integral equations 14 / 72

15 Obstacle imaging based on Helmholtz equation model Numerics Non-constant impedance λ(x) = 6 + x 1x 2 (3 + x 2 ) exact reconstructed Jijun Liu Inverse problems based on partial differential equations and integral equations 15 / 72

16 Obstacle imaging based on Helmholtz equation model Inverse scattering with mixed boundary condition Assume that D has the decomposition D = D I D D, D I D D =, where D D and D I are open curves on D. Consider the following mixed boundary value problem u + k 2 u = in R 2 \ D, u = on D D, u ν + ikσu = on D I, (5) with σ(x) := σ r (x) + iσ i (x). Jijun Liu Inverse problems based on partial differential equations and integral equations 16 / 72

17 Obstacle imaging based on Helmholtz equation model Inverse scattering problem with mixed boundary condition: Given u (, ) on S S, we need to Reconstruct the shape of the obstacle D; Reconstruct some geometrical properties of D such as normal directions and the curvature; Distinguish the coated part D I from the non-coated part D D ; Reconstruct the complex surface impedance σ(x) on D I, including the real and the imaginary parts. Jijun Liu Inverse problems based on partial differential equations and integral equations 17 / 72

18 Obstacle imaging based on Helmholtz equation model Inverse scattering with mixed boundary condition Let a D and z p a. Set Da p a C 2 -regular domain such that D Da p with z q Da p. For every fixed p, we construct three density sequences {gn p }, {fm j,p } and {h j,p l } in L 2 (S) with j = 1, 2, such that vg p n Φ(, z p ) L 2 ( Da p, n, (6) ) v fm j,p Φ(, z p ) x j, m, (7) L 2 ( Da p ) v h j,p Φ(, z p ) l x j x 2, l, (8) L 2 ( Da p ) where v f is the Hergoltz wave function with density f. Jijun Liu Inverse problems based on partial differential equations and integral equations 18 / 72

19 Obstacle imaging based on Helmholtz equation model Inverse scattering with mixed boundary condition Using these three density sequences, we construct the following three indicators I (z p ) := 1 lim γ lim u ( ˆx, d) gm(d) p gn p (ˆx) ds(ˆx)ds(d), (9) 2 m n I 1 j (z p ) := 1 γ 2 I 2 j (z p ) := 1 γ 2 lim lim m n lim where γ 2 = e π 4 i 8πk. lim S m n S S S S u ( ˆx, d) f j,p m (d) g p n (ˆx) ds(ˆx)ds(d), (1) S u ( ˆx, d) h j,p m (d) g p n (ˆx) ds(ˆx)ds(d), (11) Jijun Liu Inverse problems based on partial differential equations and integral equations 19 / 72

20 Obstacle imaging based on Helmholtz equation model Blow-up property: Using these three indicators in a different but equivalent way, we can identify the boundary property: Case 1. The geometric shape including the surface impedance is unknown. We can use the formula { lim R I +, a D I, (z p ) = (12) z p a, a D D. or Jijun Liu Inverse problems based on partial differential equations and integral equations 2 / 72

21 Obstacle imaging based on Helmholtz equation model Blow-up property: lim z p a lim z p a 2 j=1 j=1 (R I 1 j ) 2 = lim zp a [ 1 16π 2 (z p a) ν(a) 2 (κσi C(a)) ln (zp a) ν(a) 2π 2 (z p a) ν(a) 1 O( 4π (z ) = +, a D p a) ν(a) I, [ lim zp a 1 16π 2 (z p a) ν(a) 2 C(a) ln (zp a) ν(a) 8π 2 (z p a) ν(a) O( 1 4π (z p a) ν(a) ) = +, a D D ] + (κσ 2 r ) 2 (I Ij 1 ) 2 π lim 2 zp a ln 2 (z p a) ν(a) + = O(ln (z p a) ν(a) ) = +, a D I, (14) O(1), a D D ] + (1 to detect the boundary shape. We also construct the indicator functions to reconstruct the complex-valued boundary impedance, and analyze the relation between boundary curvature, boundary impedance and boundary visibility. Jijun Liu Inverse problems based on partial differential equations and integral equations 21 / 72

22 Obstacle imaging based on Helmholtz equation model Numerical performance: Example. Consider a complex obstacle D = {x : x(t) = (cos t +.65 cos 2t.65, 1.5 sin t), t [, 2π]}, A x A B E C 6 O x C C 1.5 B E A kite-shaped obstacle (left) and its curvature distribution with respect to the polar angle (right). The curvature takes maximum value near points A, B, which means a strong scattering in this part. Jijun Liu Inverse problems based on partial differential equations and integral equations 22 / 72

23 Obstacle imaging based on Helmholtz equation model For real surface impedance, the part of the boundary with the maximum curvature is relatively easy to be detected. For σ(x) 5 5i with blowup value CB =.4, the reconstruction is not improved for the part with minimum absolute value of curvature, due to the constant imaginary part. Jijun Liu Inverse problems based on partial differential equations and integral equations 23 / 72 Numerical performance Case 1. Consider the constant surface impedance for σ(x) = 5, σ(x) = 5 5i true CB=.1 CB=.2 CB=.3 CB=.4 initial.5 true CB=.1 CB=.2 CB=.3 CB=.4 initial

24 Obstacle imaging based on Helmholtz equation model Numerical performance Case 2. Curvature effect. Take σ(x) = 5 + σ i (x)i. The reconstructions with σ i (x) satisfying 1 2 C(x) + κσi (x) 5 (left) and 1 2 C(x) + κσi (x) 1 (right) in D are shown below true CB=.1 CB=.2 CB=.3 CB=.35 initial.5 true CB=.1 CB=.2 CB=.3 CB=.35 initial Jijun Liu Inverse problems based on partial differential equations and integral equations 24 / 72

25 Obstacle imaging based on Helmholtz equation model Numerical performance The uniform blowing-up property is obtained, except on the parts near the point E, where the curvature takes the negative minimum value. This phenomena is physically reasonable. There are multiple reflections of the scattered wave. So the information about this concave side is relatively small in the far-field data. To explain more about this phenomenon, a higher asymptotic expansion using higher multipole sources is needed. Jijun Liu Inverse problems based on partial differential equations and integral equations 25 / 72

26 Obstacle imaging based on Helmholtz equation model Oblique derivative conditions Takeing into account the influence of daily rotation of the Earth on the ocean waves, the diffraction of tidal waves by islands on water of constant finite depth is modeled by the following oblique derivative problem. u s + k 2 u s =, x R 2 \ D, u s ( ) u i ν + iλ us τ = ν + iλ ui, x D, τ ( u s r lim r r iku s ) =, r = x, (15) where λ is a real parameter with λ < 1. Jijun Liu Inverse problems based on partial differential equations and integral equations 26 / 72

27 Obstacle imaging based on Helmholtz equation model Oblique derivative conditions Math background: Strictly speaking, the oblique derivative should be a linear combination of normal and tangential derivatives with real coefficients, which is in fact a directional derivative of u in some direction l. Its general form is as follows: u + hu = p, (16) l where l is an unit vector, and h, p are given functions. Let µ be the tangency set consisting of the points where l is tangential to the boundary. If µ is not empty, the boundary value problem with (16) is called the Poincaré problem for which the ellipticity condition is not satisfied. Jijun Liu Inverse problems based on partial differential equations and integral equations 27 / 72

28 Obstacle imaging based on Helmholtz equation model Oblique derivative conditions Green function and reciprocity principle Let Φ s (, z) be the scattered field for point source Φ(, z). Set G(x, z) = Φ s (x, z) + Φ(x, z). Introduce (the conjugate problem of (15)) uc s + k 2 uc s =, x R 2 \ D, uc s ν iλ us c τ = p c, x D, ( ) u s lim r c iku s r c =, r = x. r Let Φ s c(x, z) be the solution to (17) with the boundary data ( ) Φ(x, z) z) p c (x) = iλ Φ(x,, x D, z R 2 \ D. ν(x) τ(x) (17) Jijun Liu Inverse problems based on partial differential equations and integral equations 28 / 72

29 Obstacle imaging based on Helmholtz equation model Oblique derivative conditions Green function and reciprocity principle Set G c (x, z) = Φ s c(x, z) + Φ(x, z), x R 2 \ D, z R 2 \ D. Theorem G(x, z) = G c (z, x) for x, z R 2 \ D with x z. Define K[ϕ](x) := 2 H[ϕ](x) := 2 D D Φ(x, y) ϕ(y)ds(y), ν(y) x D, Φ(x, y) ϕ(y)ds(y), τ(y) x D. Jijun Liu Inverse problems based on partial differential equations and integral equations 29 / 72

30 Obstacle imaging based on Helmholtz equation model Oblique derivative conditions Representation of Green function G in terms of fundamental solution Φ: Theorem For z R 2 \ D and x D, it holds that (I + iλh K) [G](x, z) = 2Φ(x, z). (18) Theorem Assume that k 2 is not a Dirichlet eigenvalue for in D. Then I + iλh K is invertible from L 2 ( D) to L 2 ( D) with a bounded inverse. Jijun Liu Inverse problems based on partial differential equations and integral equations 3 / 72

31 Obstacle imaging based on Helmholtz equation model Oblique derivative conditions Green function and reciprocity principle Theorem The far-field patterns for our scattering problem (15) and (17) have the relation u (ˆx, d) = u c ( d, ˆx), ˆx, d S. (19) Mixed reciprocity principle: Theorem For z R 2 \ D and d S, it holds that Φ c ( d, z) = σu s (z, d), σ = eiπ/4 8πk. Jijun Liu Inverse problems based on partial differential equations and integral equations 31 / 72

32 Obstacle imaging based on Helmholtz equation model Oblique derivative conditions Inverse scattering problems: IP1: Identify D from the far-field data u (ˆx, d, k) for all observation directions ˆx S, all k I and a fixed incident direction d, where S is the unit circle in R 2 and I is an interval; IP2: Identify D from the far-field data u (ˆx, d, k) for all ˆx, d S and a fixed wave number k. Jijun Liu Inverse problems based on partial differential equations and integral equations 32 / 72

33 Obstacle imaging based on Helmholtz equation model Oblique derivative conditions Inverse scattering problems: Using the analyticity of u s with respect to k, we can prove the uniqueness for IP1. Theorem The scatterer D can be uniquely determined by the far-field data u (ˆx, d, k) for all ˆx S, all k I and a fixed incident direction d. Based on the mixed reciprocity principle, we can obtain the uniqueness for IP2. Theorem Assume that D 1 and D 2 are two scatterers such that the far-field patterns u1 (ˆx, d), u 2 (ˆx, d) coincide for all observations, all incident directions and one fixed frequency. Then D 1 = D 2. Jijun Liu Inverse problems based on partial differential equations and integral equations 33 / 72

34 Obstacle imaging based on Helmholtz equation model Oblique derivative conditions Numerics for IP2 by the linear sampling method Jijun Liu Inverse problems based on partial differential equations and integral equations 34 / 72

35 Image deblurring based on fractional diffusion model The classical diffusion equation: u t = 2 u x 2. Direct diffusion: The initial value u(x, ) is smoothen as u(x, T ) at T >. The larger T is, the smoother the solution u(x, T ) is. The forward problem can be used to model the image blurring, while the backward process can be used to simulate the deblurring process. Disadvantage: The diffusion u t diffusion is required. is too strong, some slow Backward process is severely unstable. Jijun Liu Inverse problems based on partial differential equations and integral equations 35 / 72

36 Image deblurring based on fractional diffusion model The time-fractional diffusion equation: γ u t γ = u, t >, x Ω R2 for γ (, 1), Ω : the area of an image. d γ f dt γ := 1 t Γ(1 γ) Infinite propagation speed; f (s) (t s) γ ds. Sub-diffusion process, time-memory effect; Backward process is mildly unstable. Deblurring problem: u(x, ) represents the exact image with edges, can we reconstruct this image from its blurred one u(x, T ) containing some noise? Jijun Liu Inverse problems based on partial differential equations and integral equations 36 / 72

37 Image deblurring based on fractional diffusion model Backward diffusion model for image deblurring Consider the following model problem: γ u t = u, (x, t) Ω (, T ), γ u(x, ) =, (x, t) Ω (, T ), u(x, T ) = g(x), x Ω. (2) Backward problem: Approximate the distribution u(x, ) from g δ (x), the noisy measurement of g(x). Assume for some known error level δ >. Highlight paper, Inverse Problems, 213. g δ ( ) g( ) L 2 (R) δ (21) Jijun Liu Inverse problems based on partial differential equations and integral equations 37 / 72

38 Image deblurring based on fractional diffusion model Backward diffusion model for image deblurring Applying the Fourier transform with respect to x in (2), we obtain γ û(ξ, t) t γ = ξ 2 û(ξ, t). (22) Using the Laplace transform with respect to t in (22), we have û(ξ, ) = P(ξ)ĝ(ξ), P(ξ) := where E γ,1 is the Mittag-Leffler function defined by E γ,β (z) := k= 1 E γ,1 ( ξ 2 T γ ), (23) z k, z C, γ >, β >. Γ(γk + β) Jijun Liu Inverse problems based on partial differential equations and integral equations 38 / 72

39 Image deblurring based on fractional diffusion model Regularizing scheme by TV penalty term Total variation: For f L 1 (Ω; R 2 ), the total variation is defined by J TV (f ) = sup γ Q f (x) div γ(x)dx, (24) where Q = { γ = (γ 1, γ 2 ) (C 1 (Ω; R2 )) 2, γ = γ γ 2 2 1}; For f W 1,1 (Ω; R 2 ), the above definition is a weak form of J TV (f ) = f (x) dx. Ω Ω Jijun Liu Inverse problems based on partial differential equations and integral equations 39 / 72

40 Image deblurring based on fractional diffusion model Regularizing scheme by TV penalty term TV-regularization version for (23): u δ TV = arg min u A J TV (u ) such that P 1 û ĝ δ L 2 (R 2 ) δ, (25) where P 1 (ξ) = E γ,1 ( ξ 2 T γ ), A is a admissible set. Applying the Lagrangian formulation, (25) is transformed into arg min J TV (u ) + λ u A 2 P 1 û ĝ δ 2 L 2 (R 2 ), (26) where λ is a positive regularized parameter. We prove the uniqueness of the above cost functional with penalty term in frequency domain. Jijun Liu Inverse problems based on partial differential equations and integral equations 4 / 72

41 Image deblurring based on fractional diffusion model Approximation to the original minimizing problem Theorem Define two functionals J (u ) := J TV (u ) + λ 2 P 1 û ĝ 2 L 2 (R 2 ), J n (u ) := J n,tv (u ) + λ 2 P 1 n û ĝ n 2 L 2 (R 2 ), n = 1, 2, and denote by u, := arg min u A ĝ n ĝ L 2 (R 2 ), P 1 n P 1 J (u ), u,n := arg min J n (u ). If u A L 2 (R 2 ) as n and J n,tv (u ) J TV (u ) as n uniformly for all u A, then it follows that u,n u, L1 (Ω) as n. (27) Jijun Liu Inverse problems based on partial differential equations and integral equations 41 / 72

42 Image deblurring based on fractional diffusion model Important issues for the computational process: How to choose the balance parameter λ > ; How to solve the optimization problem approximately for specified λ; How to establish the error estimate for the approximate solution in terms of the noisy level δ. Bregman distance: where D p J (u, v) := J(u) J(v) p, u v, (28) p J(v) = {w : J(u) J(v) w, u v, u} is the sub-gradient of J at the point v. Jijun Liu Inverse problems based on partial differential equations and integral equations 42 / 72

43 Image deblurring based on fractional diffusion model Reconstruction scheme By introducing the Bregman distance, our optimization problem becomes u, := arg min u A with p J TV (u, ). J TV (u, )+ p, u u, +D p J TV (u, u, )+ λ 2 P 1 û ĝ δ 2 L 2 (R 2 ) Omitting the linea part p, u u, in (29), construct the Bregman iterative regularization as following: u m+1 = arg min D pm J TV (u, u m ) + λ u 2 P 1 û ĝ δ 2 L 2 (R 2 ) with p = u =. (29) := arg min Q m+1 (u) u (3) Jijun Liu Inverse problems based on partial differential equations and integral equations 43 / 72

44 Image deblurring based on fractional diffusion model Reconstruction scheme Using the definition (28), the above problem is in fact u m+1 := arg min J TV (u) J TV (u m ) p m, u u m + λ u 2 P 1 û ĝ δ 2. From the optimality of u m+1 in (3), the iteration direction in the next step can be determined by p m+1 = p m λf 1 (P 1 (P 1 û m+1 ĝ δ )). (31) Therefore (3) and (31) constitute a recursion for given initial value (u, p ) = (, ). We establish a-posterior strategy for choosing the parameters (λ, M) in terms of δ such that the solution has the error estimate: P 1 (û M û ) ( 2 + 1)δ. Jijun Liu Inverse problems based on partial differential equations and integral equations 44 / 72

45 Image deblurring based on fractional diffusion model Reconstruction scheme We now apply the fast total variation de-convolution algorithm (FTVD) to u m+1 = arg min J TV (u) + λ u 2 P 1 û ĝ m 2 (32) with ĝ m = ĝ δ + ˆf m for m =, 1,. We consider an equivalent constrained convex optimization problem: min { w + u,w Ω λ 2 P 1 û ĝ m 2 } such that w(x) = u(x). (33) Jijun Liu Inverse problems based on partial differential equations and integral equations 45 / 72

46 Image deblurring based on fractional diffusion model Reconstruction scheme We construct an iterative procedure of alternately solving a pair of easy subproblems: w-subproblem for fixed u = u : arg min Θ(w) := arg min{ w dx + β w w 2 w u 2 }. (34) u-subproblem for a fixed w = w : Ω arg min u Ψ(u) := arg min{ λ u 2 P 1 û ĝ m 2 + β 2 w u 2 }. (35) Jijun Liu Inverse problems based on partial differential equations and integral equations 46 / 72

47 Image deblurring based on fractional diffusion model Reconstruction scheme Using the definition of Frechet derivatives, the minimizer to (34) and (35) can be obtained explicitly: w[u ](x) = {, u (x) 1 β ( u (x) 1 β ) u (x) u (x), u (x) > 1 β (36) û[w, ĝ m ](ξ) = λp 1 ĝ m iβξ ŵ λp 2 + β ξ 2. (37) Jijun Liu Inverse problems based on partial differential equations and integral equations 47 / 72

48 Image deblurring based on fractional diffusion model Reconstruction scheme Algorithm TVBFD: Input: g δ, δ, λ, β, τ, tolerance, m max, k max Initialize: u =, g = g δ, p =, b =, m = While P 1 û m ĝ δ τδ or m m max u m, = u m For k =, 1,, k max w k+1 w[u m,k ] u m,k+1 u[w k+1, ĝ m ] b k+1 b k + u m,k+1 w k+1 u If m,k+1 u m,k tolerance Break u m,k+1 End For u m+1 u m,k+1 ĝ m+1 ĝ m + (ĝ δ P 1 û m+1 ) End While Output: u m+1. Jijun Liu Inverse problems based on partial differential equations and integral equations 48 / 72

49 Image deblurring based on fractional diffusion model Reconstruction scheme Example. u (x) = e x 2. Warm u(x 1,x 2,) u(x 1,x 2,1),γ=.8 u δ (x 1,x 2,1),δ=.5, n(x) / u(x,1) =.18% Neutral u δ Tikh (x 1,x,,), RelErr=158.85% u δ (x ξ 1,x 2,), RelErr=14.2% max u δ TV (x 1,x,), RelErr=3.67% 2 Cool Cold Freezing Figure: Reconstructions for different regularizing schemes (δ =.5). Jijun Liu Inverse problems based on partial differential equations and integral equations 49 / 72

50 Image deblurring based on fractional diffusion model u δ (x,x,1),δ=.5, n(x) / u(x,1) =1.77% u δ Tikh (x 1,x,), RelErr=573.81% 2 1 u δ ξ max (x 1,x 2,),RelErr=12.47% 1 u δ TV (x 1,x,), RelErr=17.23% 2 Warm Neutral u δ (x 1,x 2,1),δ=.5, n(x) / u(x,1) =17.33% u δ Tikh (x 1,x,), RelErr= % 2 u δ ξ max (x 1,x 2,), RelErr=115.84% u δ TV (x 1,x,), RelErr=27.3% 2 Cool Cold Freezing Figure: Reconstructions of different regularizing schemes, with two different noise levels δ =.5,.5. Jijun Liu Inverse problems based on partial differential equations and integral equations 5 / 72

51 Image deblurring based on fractional diffusion model Example. Consider a phantom model generated by standard function phantom.m in Matlab with default parameters. u(x 1,x 2,) u(x 1,x 2,1),γ=.8 u δ (x 1,x 2,1),δ=.5, n(x) / u(x,1) =.3% Cool Cold Freezing Nuclear Burning u δ Tikh (x 1,x,), RelErr=117.3% u δ (x,x,), RelErr=33.77% ξ 1 2 max u δ TV (x 1,x,), RelErr=4.21% 2 Hot Warm Neutral Cool Cold Freezing Figure: Reconstructions of the phantom for δ =.5. Jijun Liu Inverse problems based on partial differential equations and integral equations 51 / 72

52 Obstacle imaging based on Helmholtz equation model u(x 1,x 2,) u(x 1,x 2,1),γ=.8 u δ (x,x,1),δ=.5, n(x) / u(x,1) =.3% Cool Cold Freezing Nuclear Burning u δ Tikh (x 1,x,), RelErr=16.5% u δ (x ξ 1,x 2,), RelErr=45.83% max u δ TV (x 1,x,), RelErr=14.74% 2 Hot Warm Neutral Cool Cold Freezing Figure: Reconstructions of the phantom with the same noisy data, but the image is refined as pixels. Jijun Liu Inverse problems based on partial differential equations and integral equations 52 / 72

53 Image deblurring based on fractional diffusion model u δ (x 1,x 2,1),δ=.5, n(x) / u(x,1) =.34% 1 1 u δ Tikh (x 1,x,), RelErr=141.87% 2 1 u δ ξ max (x 1,x 2,), RelErr=48.45% 1 u δ TV (x 1,x,), RelErr=24.4% 2 Cool Cold Freezing Nuclear Burning Hot u δ (x 1,x 2,1),δ=.5, n(x) / u(x,1) =3.31% u δ Tikh (x 1,x,), RelErr=112.24% 2 u δ ξ max (x 1,x 2,), RelErr=36.21% u δ TV (x 1,x,), RelErr=54.5% Warm Neutral Cool Cold Freezing Figure: Reconstructions of the phantom for large noisy data, but the resolution keeps pixels. Jijun Liu Inverse problems based on partial differential equations and integral equations 53 / 72

54 Combustion control by absorbtion spectroscopy model Brief introduction: Purpose: Combustion control by monitoring the parameters in flame Tool: Absorbtion spectroscopy Background: Detecting the temperature and concentration of burned gas to ensure the safety of aircraft engine Difficulty: Line-of-sight (LOS), insufficient incident laser paths Mathematics: Ill-posedness, nonlinearity Jijun Liu Inverse problems based on partial differential equations and integral equations 54 / 72

55 Combustion control by absorbtion spectroscopy model Temperature and concentration detection in combustion Engineering model: Figure: The engineering configuration for the combustion (left) and its 2-dimensional cross section (right). Jijun Liu Inverse problems based on partial differential equations and integral equations 55 / 72

56 Combustion control by absorbtion spectroscopy model Absorbtion spectroscopy technique: Inject laser around some center frequency to pass through the flame; Use diode laser absorption sensors to measure the energy absorption along each laser path; Establish the nonlinear integral equation relation between temperature, concentration and laser energy Recover the temperature and concentration of burned gas from energy absorption along many laser paths Jijun Liu Inverse problems based on partial differential equations and integral equations 56 / 72

57 Combustion control by absorbtion spectroscopy model Laser injections: Figure: Laser paths distributions along perpendicular directions. Jijun Liu Inverse problems based on partial differential equations and integral equations 57 / 72

58 Combustion control by absorbtion spectroscopy model Physical law: D := {(x, y) : < x < a, < y < b}: the region of interest (ROI) The unknowns: temperature T (x, y) and concentration of burned gas X (x, y). P: the constant gas pressure in D; γ: the injected laser frequency distributed in [γi 1, γ2 i ] with center frequency γ i ; S(T (x, y), γ i ): the laser strength for injected laser with center frequency γ i, which has the approximate expression S(T, γ i ) = S(T, γ i ) Q(T ) Q(T ) exp [ hce i ( 1 k T 1 ) T φ(p, T (x, y), γ γ i ): the laser function satisfying ]. (38) γ 2 i γ 1 i φ(p, T (x, y), γ γ i )dγ = 1. (39) Jijun Liu Inverse problems based on partial differential equations and integral equations 58 / 72

59 Combustion control by absorbtion spectroscopy model Parameters detections by nonlinear integral equation Nonlinear integral equation For injected laser along perpendicular directions, we have integral equation systems with respect to X (x, y) and T (x, y): A(y, γ i ) = P B(x, γ i ) = P a b X (x, y )S(T (x, y ), γ i )dx, y (, b). (4) X (x, y)s(t (x, y), γ i )dy, x (, a). (41) Problem: For given measurement data {(A(y, γ i ), B(x, γ i ))} for all x (, a), y (, b) and several center frequencies γ i, identify (X (x, y), T (x, y)) in D by the model (4)-(41). Jijun Liu Inverse problems based on partial differential equations and integral equations 59 / 72

60 Combustion control by absorbtion spectroscopy model Parameters detections by nonlinear integral equation Engineers scheme L ( min (X,T ) i=1 a X (x, )S(T (x, ), γ i )dx A(, γ i ) 2 L 2 (,b) + b ) X (, y)s(t (, y), γ i )dy B(, γ i ) 2 L 2 (,a). Firstly discretize the integrals in finite dimensional space, then solve this nonlinear optimization problem by iteration scheme. Disadvantages: Large memory required: 2 N N-unknowns optimization; Convergence not guaranteed (nonconvex functional, multiple local minizers) Time consuming (Spend three hours for N = 2 and get useless result.) Jijun Liu Inverse problems based on partial differential equations and integral equations 6 / 72

61 Combustion control by absorbtion spectroscopy model Parameters detections by nonlinear integral equation Mathematical observations: Although the system (4)-(41) is nonlinear with respect to (X, T ), it becomes a linear system with respect to X (x, y) if T (x, y) is given. The linear system (4)-(41) with respect to X (x, y) for given T (x, y) is unstable, i.e., a small perturbation in (A, B) may cause large change in the solution X (x, y); For given X (x, y), the determination of T (x, y) from is also a nonlinear ill-posed problem. Jijun Liu Inverse problems based on partial differential equations and integral equations 61 / 72

62 Combustion control by absorbtion spectroscopy model Semilinear alternative iteration schemes Assume (X (x, y), T (x, y)) is the exact unknown parameters. Given a-priori approximate value ( X (x, y), T (x, y)) of (X (x, y), T (x, y)) and the initial iteration value (X (x, y), T (x, y)); Given the regularizing parameters α, β > ; iteration stopping level ε > ; For known T n (x, y), generate X n+1 (x, y) from the regularizing solution of the linear ill-posed system { (K n a X )(y) := a X (x, y)s(t n (x, y), γ i )dx = A(y, γ i ), < y < b (K n b X )(x) := b X (x, y)s(t n (x, y), γ i )dy = B(x, γ i ), < x < a, that is, X n+1 := arg min X { } K n a X A 2 + K n b X B 2 + α X X 2 L 2 (D). (43) (42 Jijun Liu Inverse problems based on partial differential equations and integral equations 62 / 72

63 Combustion control by absorbtion spectroscopy model Semilinear alternative iteration schemes For known X n+1 (x, y), solving T n+1 (x, y) from the regularizing solution for the nonlinear ill-posed system { (G n+1 a T )(y) := a (G n+1 b that is, T n+1 := arg min T X n+1 (x, y)s(t (x, y), γ i )dx = A(y, γ i ), < y < b T )(x) := b X n+1 (x, y)s(t (x, y), γ i )dy = B(x, γ i ), < x < a, { G n+1 a T A 2 + G n+1 b T B 2 + β T T } 2 L 2 (D) (45) If (X n+1, T n+1 ) (X n, T n ) L 2 (D) ε, stop and take (X n+1, T n+1 ) as the approximate solution; otherwise, continue the iteration. Jijun Liu Inverse problems based on partial differential equations and integral equations 63 / 72

64 Combustion control by absorbtion spectroscopy model Semilinear alternative iteration schemes To improve performance, apply multiple center frequencies for the injected lasers at each direction. That is, (43) and (45) are replaced by { L } X n+1 [ := arg min K n a,i X A i 2 + K n b,i X B i 2] + α X X 2, X i=1 { (46) L } [ T n+1 := arg min G n+1 a,i T A i 2 + G n+1 b,i T B i 2] + β T T 2. T i=1 (47) On the choice strategy for parameters α > and β >. Jijun Liu Inverse problems based on partial differential equations and integral equations 64 / 72

65 Combustion control by absorbtion spectroscopy model Numerical implementations Example. (X1, T 1 ) with unimodal distributions: T1 (x.5)2 (x, y) : = ( ( (x.5) (y.5) e X1 (x.5)2 (x, y) : = ( e ) ( (x.5) (y.25) ) ) (y.5) ) (y.5) (48) Jijun Liu Inverse problems based on partial differential equations and integral equations 65 / 72

66 Combustion control by absorbtion spectroscopy model Numerical implementations Real T inital T Real X inital X Figure: The unimodal distribution together with the initial values for iterations. Jijun Liu Inverse problems based on partial differential equations and integral equations 66 / 72

67 Combustion control by absorbtion spectroscopy model Numerical implementations Real T recon T error T Real X recon X error X Figure: The reconstruction performance together with the error distributions. Jijun Liu Inverse problems based on partial differential equations and integral equations 67 / 72

68 Combustion control by absorbtion spectroscopy model Numerical implementations Example. both T2 (x, y) and X 2 (x, y) have two peaks, see Figure 9 (left) for the exact distribution (X 2, T 2 ) := { (X1 (x, y), T 1 (x, y)), x 1, y.5, (X1 (x, y.5), T 1 (49) (x, y.5)), x 1,.5 < y 1. Jijun Liu Inverse problems based on partial differential equations and integral equations 68 / 72

69 Combustion control by absorbtion spectroscopy model Numerical implementations Figure: The bimodal distribution together with the initial values for iterations. Jijun Liu Inverse problems based on partial differential equations and integral equations 69 / 72

70 Combustion control by absorbtion spectroscopy model Numerical implementations Real T recon T error T Real X recon X error X Figure: The reconstruction performance together with the error distributions. Jijun Liu Inverse problems based on partial differential equations and integral equations 7 / 72

71 Research Group at Southeast University: Research Institutes at SEU: School of Information Science and Engineering School of Biological Sciences and Medical Engineering School of Energy and Environment Department of Mathematics Jijun Liu Inverse problems based on partial differential equations and integral equations 71 / 72

72 Thank you for your attention! Jijun Liu Inverse problems based on partial differential equations and integral equations 72 / 72